1. Field of the Invention
The present invention relates to communications systems and, more particularly, to phase angle demodulators.
2. Description of the Related Technology
Most electronic communication systems in use today include a transmitter to transmit an electromagnetic signal and a receiver to receive the transmitted signal. The transmitted signal is typically corrupted by noise and, therefore, the receiver must operate with received data that reflects the combination of the transmitted signal and noise. Thus, the receiver receives data y(t) at a time t, where y(t)=s(t)+n(t), the sum of the transmitted signal and additive noise. The received data equation can be expanded as follows: EQU y(t)=.sqroot.2Acos[.omega..sub.o t+.theta.(t)]+n(t) (1)
where A is the signal amplitude, .omega..sub.o is the carrier or reference frequency, .theta.(t) is the time-varying phase function and n(t) is noise.
Many of these communication systems require that the receiver demodulate information in the received signal which depends on proper demodulation of the signal phase angle at all times during transmission. The demodulation of the signal phase angle is problematic in view of the pervasiveness of noise. Therefore, for this class of receivers it is desirable to optimize phase demodulation, which is equivalent to optimizing an estimation of the phase function .theta.(t).
Among the class of receivers which rely on accurate phase demodulation are current high quality frequency modulated (FM) receivers. Such FM receivers typically use phase-locked loops, or PLLs, for phase demodulation. A phase-locked loop is a circuit that consists of a phase detector which compares a frequency of a voltage-controlled oscillator (VCO) with that of an incoming carrier signal. A phase-error signal output by the phase detector, after passing through a linear filter, is fed back to the voltage-controlled oscillator to keep the oscillator generated frequency "locked" in a fixed phase relationship with the input or reference frequency. The phase-error signal output of the linear filter is a low frequency (baseband) signal that is proportional to the input frequency and, thus, represents the demodulated information in the FM signal.
Systems incorporating phase-locked loops are suboptimal since the phase-locked loop is a "causal" system. A system is causal if the output at any given time depends only on values of the input at the present time and in the past. Such a system can also be referred to as "nonanticipative", as the system output does not anticipate future values of the input.
Ideally, then, a noncausal receiver makes an estimate of the phase function .theta.(t), given phase samples .theta.(k.DELTA.), k=1,2,. . .,K where .DELTA. is a sampling period, in an optimal manner known as maximum a posteriori (MAP) estimation. Such phase samples based on noisy data can be measured by the extraction of in-phase (I) and quadrature (Q) components from the data which determine a measured phase angle according to the arctangent operation tan.sup.-1 (Q/I).
The optimization of phase demodulation can then be expressed as minimizing the mean-squared error between the phase estimate .theta.(.tau.) and the correct phase value .theta.(.tau.), with extra information provided by the mean phase .theta..sub.m (.tau.) of the prior (a priori) phase distribution. This approach implies that the receiver must solve a nonlinear integral equation as follows: ##EQU1## where .theta.(.tau.)-.theta..sub.m (.tau.) is the difference between the phase estimate .theta.(.tau.), at a time .tau., t.sub.o .ltoreq..tau..ltoreq.t, and the prior mean phase .theta..sub.m (.tau.) at time .tau., N.sub.O /2 is the noise power spectral density, .sigma. is time, and R.sub..theta. (.tau.-.sigma.) is the phase covariance function defined as the expected value E{[.theta.(.tau.)-.theta..sub.m (.tau.)] [.theta.(.sigma.)-.theta..sub.m (.sigma.)]}. If sampled values of .theta.(.tau.) are used to form a column vector .theta., then a sampled version of R.sub..theta. (.tau.-.sigma.) can be obtained from the phase covariance matrix E[(.theta.-.theta..sub.m)(.theta.-.theta..sub.m).sup.T ].
The use of a phase-locked loop (PLL) for phase estimation, as for example shown in FIG. 1, follows from the similarity of the loop equation, equation (3) below, to the estimate in equation (2). The loop equation, which is implemented as a phase-locked loop, is described by the following equation: ##EQU2## where f(.tau.) is the impulse response of the linear filter in the PLL. Equations (2) and (3) are similar if .theta..sub.m (t) is either 0 or is added to the estimate obtained from the loop equation. Although the equations are superficially similar, they differ in that whereas the phase-locked loop uses information in the interval [-.infin.,.tau.], the optimum demodulator uses all of the information in the observation integral [t.sub.0,t] to determine the phase estimate at time .tau..
The optimum estimation process is thus noncausal. That is, "future" samples of the signal in the interval [.tau., t], from the MAP estimation (Equation (2)), are used to determine the phase at a given time. On the other hand, the phase locked-loop is a realizable but suboptimal version of the ideal estimator due to its inherent causality.
Another shortcoming of the phase-locked loop is that it is difficult to incorporate a time-varying signal amplitude A(t) and/or time-varying noise power .sigma..sub.n.sup.2 (t)=E[y(t)-s(t)].sup.2, i.e., the variance between the received data and the transmitted signal. There are many non-ideal environments in which such variations occur. In fact, fading (signal amplitude fluctuations) and time-dependent noise power are phenomena that are prevalent in mobile receivers such as aircraft and automobiles.
D. W. Tufts and J. T. Francis proposed a numerical method to more closely approach the performance of an optimal MAP phase angle estimator by using a combination of block processing and a priori phase information obtained from past data samples ("Maximum Posterior probability Demodulation of Angle-Modulated Signals", IEEE Transactions on Aerospace Electronic Systems, AES-15 (1979), pp. 219-227). Instead of processing samples one at a time, the Tufts and Francis block process uses a sequence of samples from a time interval defined by a block. An estimated phase sample at the beginning of the interval can thus be influenced by data samples at the end of the interval, emulating the desired noncausal process.
In such a block process, the demodulated phase samples are available after the complete block has been processed. This implies that there is a delay of up to one block length between a data sample and the corresponding phase estimate. Nonetheless, such a delay can be so small as to be unnoticeable in a two-way communication system, and is irrelevant to one-way systems such as consumer radios, televisions receivers and facsimile machines.
For block processing in linear signal estimators, the error of a time invariant signal can be reduced by averaging noisy samples over the duration of the block. For linear estimation of a time-varying signal, the averaging process is replaced by filtering, where the filter is based on the expected correlation between signal samples. The same correlation information can be used to predict a given sample from past data, and a MAP estimator often forms a weighted sum of a prediction based on past data and filtered, demodulated samples from the current data block.
For phase estimation, the nonlinear division and arctangent operation over quadrature components (I, Q) necessitate a more complicated nonlinear filtering process in order to combine a block of measured, noisy phase samples .alpha.(j.DELTA.) so as to reduce the phase-error of each sample. The discrete time equation for the MAP phase estimate at a time k.DELTA. is as follows: ##EQU3## Equation (4) is obtained from equation (2) by defining y(j.DELTA.)=.vertline.y(j.DELTA.).vertline.cos[.omega..sub.0 j.DELTA.+.alpha.(j.DELTA.)]. Tufts and Francis described the iterative method to solve equation (4) (given in equation (5) below).
The phase estimation process uses a weighted sum of a phase sample .theta..sub.m (k.DELTA.) predicted from past data and a processed version of the measured phase samples .alpha.(j.DELTA.) from the current data block. However, the operation that must be applied to current phase samples is difficult to implement because the desired MAP phase estimate .theta.(k.DELTA.) from the current data block is a filtered version of a nonlinear (sine function) version of the same desired phase estimate, as well as desired estimates at other sampling times within the data block. Thus, because of the processing time and complexity involved, the iterative MAP estimation method is inefficient for practical, real-time receivers which employ standard, sequential instruction execution, or von Neumann, computers.
Recent innovations in parallel distributed processing based on fundamental biological notions about the human brain present an alternative to von Neumann computers. These so-called neural networks have been demonstrated to solve, or closely approximate, very difficult nonlinear optimizations. One of the original neural networks, as disclosed in Hopfield (U.S. Pat. No. 4,660,166) is a crossbar network of operational amplifiers which is shown to be used as an associative memory. A subsequent patent to Hopfield, et al. (U.S. Pat. No. 4,719,591) incorporates a second interconnection network for decomposing an input signal comprising one or more input voltages, in terms of a selected set of basis functions.
As defined herein in the present disclosure, a Hopfield network (also called a crossbar network) comprises a set of amplifiers, typically including operational amplifiers, that are interconnected by feedback lines. The output of each operational amplifier is nonlinearly filtered so that the output voltage of each amplifier lies in the unit interval [0,1]. Network stability is assured because the maximum voltage at each output is thereby limited.
The network amplifiers are fed by a set of bias currents, {b}, which are generated external to the network optional configuration, a set of variable gain amplifiers are interposed in the feedback lines. The gain on each amplifier is adjusted by a set of feedback weights, {T}. Thus, given input comprising bias currents and feedback weights, the network will eventually converge to a stable state in which the output voltages of the amplifiers remain constant and represent a solution to the specific input data and application. Hopfield networks have been applied to receivers such as those of Vallet (U.S. Pat. No. 4,656,648) and Provence (U.S. Pat. No. 4,885,757) but none have used a Hopfield network for phase angle demodulation, and none have used the specific nonlinearity required for such an application.
In summary, phase angle demodulation is fundamental to many communication systems and FM radio receivers. Currently, high quality FM receivers use phase-locked loops for demodulation, but these systems are theoretically suboptimal. The optimal demodulator must solve a nonlinear integral equation. As is well known, such an integral solution is difficult to achieve in real-time. If a simple analog or digital circuit could be found to more closely approximate the desired optimal demodulator, such a circuit would improve demodulation performance thereby replacing the traditional phase-locked loop found in high performance radio, television and communication systems. It would be an additional benefit if a signal detector-classifier could make use of such a proposed circuit to thereby correlate the estimated phase of the received signal against the phase of a synthesized signal vis-a-vis the standard linear Wiener or Kalman filter methods of correlating the signals themselves.